Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,32,38)(27,33,41,49)(28,34,46,54)(29,35,31,37)(30,36,40,48)(39,47,55,56)(42,50,44,52)(43,51,45,53);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,32,41)(26,33,38,49)(28,42,46,44)(29,40,31,30)(34,50,54,52)(35,48,37,36)(39,43,55,45)(47,51,56,53);(2,5)(6,8)(25,28,29,43,32,46,31,45)(26,34,35,51,38,54,37,53)(27,39,30,44,41,55,40,42)(33,47,36,52,49,56,48,50);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,32,31)(26,35,38,37)(27,40,41,30)(28,43,46,45)(33,48,49,36)(34,51,54,53)(39,42,55,44)(47,50,56,52);(1,3,4,7)(2,6,5,8)(11,19)(14,17)(18,24)(22,23)(25,30,31,27,32,40,29,41)(26,36,37,33,38,48,35,49)(28,44,45,39,46,42,43,55)(34,52,53,47,54,50,51,56)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=hi,d^2=fi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6738447);